Consider the heat equation $$ u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1} $$ for a Hölder continuous coefficient $a(x,t)$ satisfying $$ 0<C_o \le a(x,t) \le C_1. $$

Now $u$ satisfies the Harnack inequality $$ \sup_{Q^-(r^2,r)} u \le H_o \inf_{Q^+(r^2,r)}u $$ for a constant $H_o = H_o(d, C_o, C_1)>0$, and $$ Q^-(r^2,r) :=B(r/2) \times \left[-\frac{3r^2}4, -\frac{r^2}2\right]\quad and\quad Q^+(r^2,r) :=B(r/2) \times \left[-\frac{r^2}4, 0\right]. $$

My question is that what is the dependence of $H_o$ on $C_o$ and $C_1$. In particular, I am interested in knowing what happens to $H_o$ if $C_o$ and $C_1$ are replaced by $AC_o$ and $AC_1$ for $A\ge 1$. This can, of course, be answered by just going through the proof and checking what happens, but this is quite tedious and I was wondering if there is a simpler way to see this and/or if someone knows this by heart.

Another way to put this is to consider the equation with a scaled time $s=t/A$, which cancels the constant $A$ from the structure conditions, but then again, this also requires scaling the sets in the Harnack inequality, so it does not seem completely obvious what happens to the constant.